Ikkeabeliansk

Ikkeabeliansk is a term used in mathematics to describe a non-abelian structure, most commonly a group whose binary operation is not commutative. In other words, there exist elements a and b such that ab does not equal ba. If ab = ba for all pairs of elements, the structure is abelian.

In group theory, non-abelian groups are those in which the order of applying two elements matters. Classic examples include the symmetric group S3, the dihedral group D4 (the symmetries of a square), and the quaternion group Q8. In these groups, one can find pairs of elements a and b for which ab ≠ ba, illustrating the departure from commutativity. Notably, many groups with multiple generators and nontrivial relations are non-abelian, and the symmetric groups S_n are non-abelian for all n ≥ 3.

The non-abelian property has important implications. It leads to the study of commutators, conjugation, and the derived subgroup, as well as the process of forming the abelianization of a group, which is the largest abelian quotient of the original group. This structure also appears in various mathematical areas and in physics, where non-abelian gauge theories play a central role in describing fundamental interactions.

In brief, ikkeabeliansk describes systems where the operation is inherently non-commutative, a feature that gives rise to rich theory and diverse applications. See also non-abelian group and abelianization.